High School Optimal Final Jeopardy Strategy: All In or Zero?

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SUMMARY

The optimal strategy for Final Jeopardy, particularly in a tie scenario, is to either bet all in or nothing at all. In a recent discussion, a contestant with $1,000 won after the two contestants with $12,000 each bet their entire stakes and answered incorrectly. This outcome challenges the conventional wisdom that suggests betting slightly less than the maximum. The analysis indicates that understanding the probabilities of each contestant's correct answers and their betting behaviors is crucial for determining the best strategy.

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Hornbein
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TL;DR
Calculating the best Final Jeopardy strategy.
Once in Final Jeopardy the first contestant held $1000 with the other two tied with $12,000 apiece. The first bet $999 and got the question wrong, leaving him with one dollar. The other two both bet their entire stake and got the answer wrong, leaving them with zero dollars. The first contestant won.

I opined that the two bet correctly. Their only reasonable bets were either all in or zero. It seems though that this goes against the popular wisdom of Jeopardy fans, who claimed that betting slightly less than the maximum was the way to go. Perhaps this is true in general but I maintain it false in the case of a tie.

One can complicate things by guessing the likelyhood that each contestant will be able to come up with the correct answer and making a payoff matrix, but such estimates are so speculative I think this isn't worth the bother.
 
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In game theory, the optimal answer often depends on the policy of your opponents. Usually, the best policy includes some random behavior so that your opponents are not sure what you will do (and you are not sure of what they will do). All anyone can know are the probabilities.
 
Hornbein said:
Summary:: Calculating the best Final Jeopardy strategy.

One can complicate things by guessing the likelyhood that each contestant will be able to come up with the correct answer and making a payoff matrix, but such estimates are so speculative I think this isn't worth the bother.
Part of the reason that they reveal the category of the final jeopardy question before the contestants make their wagers is to let the contestants do just that. Do you remember what the category was for this question?
 
berkeman said:
Part of the reason that they reveal the category of the final jeopardy question before the contestants make their wagers is to let the contestants do just that. Do you remember what the category was for this question?
It was Asian Geography.
 
Yes, guessing the likely hood of each contestant if they will bet it all or just shy of it will require you to look at their bets on the daily doubles. Do they bet it all or just shy of it? Make sense?
 
This is an interesting optimization question and is (I think) subject to exact (optimal: best expected outcome for repeated play) solution.
Question: what must be known to provide the optimal solution?
My answer:
  1. Probability of each contestant getting the correct answer
  2. Probability of each contestant making each particular wager
Is that correct (my apologies for this small hijacking)
 
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3. How bad would I feel if I bet nothing (based on the category) and the it turned out I knew the question after all?

Category is Shakespeare!

gmax137: $0, Hi Mom!

The Answer is: "She is Caliban's Mother"

gmax137: "Sycorax, dammit! The only Shakespeare I remember; we read "The Tempest" in 7th grade :H :H :frown:
 
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gmax137 said:
3. How bad would I feel if I bet nothing (based on the category) and the it turned out I knew the question after all?

Category is Shakespeare!
Me, with anything sports. I usually make negative bets. ("I bet $10,000 I don't know this one.")
 
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hutchphd said:
This is an interesting optimization question and is (I think) subject to exact (optimal: best expected outcome for repeated play) solution.
Question: what must be known to provide the optimal solution?
My answer:
  1. Probability of each contestant getting the correct answer
  2. Probability of each contestant making each particular wager
Is that correct (my apologies for this small hijacking)
Edit: You'd imagine , unless you know how well the two players know about Asian Geography from another question in the game, that , given that the players earned $12,000, that their knowledge is overall well-rounded, so that the probability of getting the question right is reasonably-high. Of course, you can make the estimate more precise, using the details and your observations about this specific game, but you'll likely be busy -enough with the game to use all your computing power towards answering questions.
 
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WWGD said:
but you'll likely be busy -enough with the game to use all your computing power towards answering questions.
I'm pretty sure they're both pre-coached and game-assisted to make strategically shrewd bets.
 

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